Given that it's a new business, I would agree with the group that wants to go with Server Provider. See the explanation below.
Hosting one's website personally can be quite expensive. Think of this in terms of:
Actual Financial Cost of Acquiring the relevant equipment. To keep up with the growth of the business, you will need to incur the cost of depreciation and maintenance as well.
Cost of Possible liability due to loss of data:
Should anything happen to the data of your clients, the cost may be too much to bear.
Shared hosting on the other hand has the following advantages:
Hence, I'd go with Server Provider.
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